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Here the 'IEEE 754 double value' resulting of the 15 bit figure is 3.330560653658221E-15, which is rounded by Excel for the 'user interface' to 15 digits 3.33056065365822E-15, and then displayed with 30 decimals digits gets one 'fake zero' added, thus the 'binary' and 'decimal' values in the sample are identical only in display, the values ...
There are two common rounding rules, round-by-chop and round-to-nearest. The IEEE standard uses round-to-nearest. Round-by-chop: The base-expansion of is truncated after the ()-th digit. This rounding rule is biased because it always moves the result toward zero.
In the example from "Double rounding" section, rounding 9.46 to one decimal gives 9.4, which rounding to integer in turn gives 9. With binary arithmetic, this rounding is also called "round to odd" (not to be confused with "round half to odd"). For example, when rounding to 1/4 (0.01 in binary), x = 2.0 ⇒ result is 2 (10.00 in binary)
C# has a built-in data type decimal consisting of 128 bits resulting in 28–29 significant digits. It has an approximate range of ±1.0 × 10 −28 to ±7.9228 × 10 28. [1] Starting with Python 2.4, Python's standard library includes a Decimal class in the module decimal. [2] Ruby's standard library includes a BigDecimal class in the module ...
After padding the second number (i.e., ) with two s, the bit after is the guard digit, and the bit after is the round digit. The result after rounding is 2.37 {\displaystyle 2.37} as opposed to 2.36 {\displaystyle 2.36} , without the extra bits (guard and round bits), i.e., by considering only 0.02 + 2.34 = 2.36 {\displaystyle 0.02+2.34=2.36} .
Decimal arithmetic, compatible with that used in Java, C#, PL/I, COBOL, Python, REXX, etc., is also defined in this section. In general, decimal arithmetic follows the same rules as binary arithmetic (results are correctly rounded, and so on), with additional rules that define the exponent of a result (more than one is possible in many cases).
A round number is mathematically defined as an integer which is the product of a considerable number of comparatively small factors [12] [13] as compared to its neighboring numbers, such as 24 = 2 × 2 × 2 × 3 (4 factors, as opposed to 3 factors for 27; 2 factors for 21, 22, 25, and 26; and 1 factor for 23).
For example, the decimal number 123456789 cannot be exactly represented if only eight decimal digits of precision are available (it would be rounded to one of the two straddling representable values, 12345678 × 10 1 or 12345679 × 10 1), the same applies to non-terminating digits (. 5 to be rounded to either .55555555 or .55555556).